Page 232 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The model in Eqn 12.2 can be expanded to include epistatic effects as:
y = Xb + Za + Wd + Sep + e
where ep is the vector of interaction (epistatic) effects. The evaluation can be carried
out as described in Section 12.3 but the major limitation is obtaining the inverse of the
epistatic relationship matrix for large data sets. However, VanRaden and Hoeschele
(1991) presented a rapid method for obtaining the inverse of the epistatic relationship
matrix when epistasis results from interactions between additive by additive (A × A)
genetic effects when the population is inbred or not. The approach is similar to the
method described for obtaining the inverse of the dominance relationship matrix and
it involves including sire × dam subclasses; consequently, the details of the method have
not been covered in this section. The method involves calculating the inverse of U, the
−1
relationship matrix among epistatic and subclass effects, and U is then included in
the usual MME for the prediction of epistatic and sire × dam subclass effects.
−1
The rules for obtaining U for a population that is not inbred are given in the
next section, with an illustration.

12.5.1 Rules for the inverse of the relationship matrix for epistatic
and subclass effects

The inverse of U can be computed by going through a list of individuals and their
parents and sire × dam subclasses. See rules 1 and 2 in Section 12.4.1 on how such a
list should be set up. The contribution of individual i in the list to U is computed
−1
by the following rules:
1. For an individual i with both parents and subclass effects known, the contribution
−1
to U is:
c s d ( s d)
,
⎡ 16 −4 −4 −16 ⎤
⎢ ⎥
⎢ −4 1 1 4 ⎥ / (112 ) (12.16)
⎢ −4 1 1 4 ⎥
⎢ ⎥
⎣ −16 4 4 16 ⎦

2. For an individual with both parents known but subclass effects treated as
−1
unknown, the contribution to U is:
c s d
⎡ 16 − 4 − 4⎤
⎥
⎢ − 1 114)
⎢ 4 1 ⎥ ( / (12.17)
⎢ ⎣ − 4 1 1⎥ ⎦
3. If only one parent, say s, is known, then the contribution is:
c s
⎡ 16 − 4⎤ 115)
⎢ ⎥ ( / (12.18)
⎣ − 4 1 ⎦

216 Chapter 12